In this week’s homework, we’ll go through a few common problems that students have with the exogenous growth model. Do not spend more than 6-7 solid hours’ work on them. These are all classic exam questions.
Statement of problem:
In an economy, aggregate production is produced according to the Cobb Douglas production function:
\[ Y_{t} = K_{t}^{\alpha}(A_{t}L_{t})^{1-\alpha} \]
where \(Y\) is output, \(K\) is the capital stock, \(L\) is the labour force, and \(A_{t}\) is a Hicks-neutral technology. The labour force grows according to \(L_{t+1} = L_{t}(1+n)\), and technology grows at \(A_{t+1} = A_{t}(1+g)\). The capital stock evolves according to \(K_{t+1} = (1-\delta)K_{t} + I_{t}\), where \(\delta\) is the depreciation rate and \(I=sY\) is investment, with \(s\) being an exogenously determined savings rate. Firms in the economy maximise profits
\[ \Pi = K_{t}^{\alpha}(A_{t}L_{t})^{1-\alpha} - wL-rK \]
with \(w\) being the prevailing wage rate and \(r\) being the cost of capital.
When firms are using the amount of capital that maximises profits, how much capital is used? What is the return on capital?
Repeat the above, but for labour. If all income \(Y\) is either labour income \(wL\) or capital income \(rK\), what is the relationship between the parameters of the production function and the capital/labour shares?
Another common production function is the “CES” production function, which takes the form
\[ Y = \left(\alpha K^{\rho} +(1 - \alpha)L^{\rho} \right)^{\frac{1}{\rho}} \]
Derive the marginal products of capital and labour for this form. Hint: apply the chain rule.
Express the Cobb-Douglas version of the model in per-effective worker terms (both production function and capital accumulation equation). That is, divide both parts of the model by \(AL\). Call the resulting capital and output per effective units of labour \(k\) and \(l\).
Under balanced growth, the capital-output ratio is constant. Express the balanced growth path of \(y^{*}\) and \(k^{*}\) in terms of the exogenous variables. How quickly are capital and output growing during balanced growth?
We don’t always assume that economies are at the equilibrium point; more, it is an attracting point. During WWII, much of Germany’s capital stock was destroyed, though technology was not. Illustrate what happened to Germany during the post-war years on a Solow-Swan diagram. Tell a story!
Finally, we’re not interested in just finding the balanced growth path; we want to find the one that maximises some measure of wellbeing. One measure of wellbeing is the dollar value of material goods that we purchase.
Plot the savings rate (x axis) against the amount of consumption in the economy (y axis). What is the shape? If a country wanted to maximise its consumption, what rate would you recommend?